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In this work, we study the existence of $W_0^{1, p(\cdot)}$-solutions to thefollowing boundary value problem involving the $p(\cdot)$-Laplacian operator:  \begin{equation*}  \left\lbrace  \begin{array}{l}  -\Delta_{p(x)}u+|\nabla u|^{q(x)}=\lambda g(x)u^{\eta(x)}+f(x),\quad\textnormal{ in } \Omega, \\\qquad \,\,\,\,\,\quad \quad\qquad\quad u\geq0, \quad\textnormal{ in } \Omega  \qquad \,\,\,\,\,\quad \quad\qquad\quad u= 0, \,\,\quad \text{on }\partial\Omega.  \end{array}  \right. \end{equation*}under appropriate ranges on the variable exponents. Wegive assumptions on $f$ and $g$ in terms of the growth exponents $q$ and $\eta$under which the above problem has a non-negative solution for all $\lambda >0$.

Effect of Non-linear Lower Order Terms in Quasilinear Equations Involving the $p(x)$-Laplacian